Non-Lipschitz homeomorphism from compact metric space to itself

Question

Is it possible to find a compact metric space $(X,d)$ with more than one point and a homeomorphism $\varphi:(X,\tau) \to (X,\tau)$ where $\tau$ is the topology induced by $d$ such that $$(\forall N\in \mathbb{N})(\exists x, y\in X): x\neq y \text{ and } d(f(x),f(y)) \geq N\cdot d(x,y)?$$

Answer 1

$X=[0,1]$ with the usual metric, $\phi(x)=\sqrt x$.